Answer :
Let's break this down step by step.
### Part 1: Finding the smallest number whose prime factors are 2, 3, 5, and 11
To find the smallest number whose prime factors are 2, 3, 5, and 11, we simply multiply these prime numbers together:
[tex]\[ 2 \times 3 \times 5 \times 11 \][/tex]
Calculating the product:
[tex]\[ 2 \times 3 = 6 \][/tex]
[tex]\[ 6 \times 5 = 30 \][/tex]
[tex]\[ 30 \times 11 = 330 \][/tex]
So, the smallest number whose prime factors are 2, 3, 5, and 11 is:
Answer: (iv) 330
### Part 2: Prime Factorization of Given Numbers
#### (a) 90
Let's break down 90 using a prime factorization tree:
1. Divide by the smallest prime number (2):
[tex]\[ 90 \div 2 = 45 \][/tex]
2. Divide 45 by the next smallest prime number (3):
[tex]\[ 45 \div 3 = 15 \][/tex]
3. Continue with 3:
[tex]\[ 15 \div 3 = 5 \][/tex]
4. Finally, 5 is already a prime number:
[tex]\[ 5 \div 5 = 1 \][/tex]
So, the prime factorization of 90 is:
[tex]\[ 90 = 2 \times 3 \times 3 \times 5 \][/tex]
or in exponential form:
[tex]\[ 90 = 2^1 \times 3^2 \times 5^1 \][/tex]
#### (b) 749
1. Checking the divisibility by primes, you find that 749 is divisible by 7:
[tex]\[ 749 \div 7 = 107 \][/tex]
2. Next, checking 107, it's a prime number.
So, the prime factorization of 749 is:
[tex]\[ 749 = 7 \times 107 \][/tex]
#### (c) 625
1. Divide by the smallest prime number (5), since 625 ends with a 5:
[tex]\[ 625 \div 5 = 125 \][/tex]
2. Continue dividing by 5:
[tex]\[ 125 \div 5 = 25 \][/tex]
[tex]\[ 25 \div 5 = 5 \][/tex]
[tex]\[ 5 \div 5 = 1 \][/tex]
So, the prime factorization of 625 is:
[tex]\[ 625 = 5 \times 5 \times 5 \times 5 \][/tex]
or in exponential form:
[tex]\[ 625 = 5^4 \][/tex]
#### (d) 1480
1. Divide by the smallest prime number (2):
[tex]\[ 1480 \div 2 = 740 \][/tex]
[tex]\[ 740 \div 2 = 370 \][/tex]
[tex]\[ 370 \div 2 = 185 \][/tex]
2. Divide by the next smallest prime number (5):
[tex]\[ 185 \div 5 = 37 \][/tex]
3. Finally, 37 is a prime number.
So, the prime factorization of 1480 is:
[tex]\[ 1480 = 2 \times 2 \times 2 \times 5 \times 37 \][/tex]
or in exponential form:
[tex]\[ 1480 = 2^3 \times 5^1 \times 37^1 \][/tex]
#### (e) 6182
1. Divide by the smallest prime number (2):
[tex]\[ 6182 \div 2 = 3091 \][/tex]
2. Continue checking for prime factors:
3091 is not divisible by 2, 3, or 5. Checking for larger primes, we find:
[tex]\[ 3091 \div 11 = 281 \][/tex]
3. 281 is a prime number.
So, the prime factorization of 6182 is:
[tex]\[ 6182 = 2 \times 11 \times 281 \][/tex]
### Summary
- The smallest number whose prime factors are 2, 3, 5, and 11 is 330.
- Prime factorizations:
- 90: [tex]\( 2^1 \times 3^2 \times 5^1 \)[/tex]
- 749: [tex]\( 7^1 \times 107^1 \)[/tex]
- 625: [tex]\( 5^4 \)[/tex]
- 1480: [tex]\( 2^3 \times 5^1 \times 37^1 \)[/tex]
- 6182: [tex]\( 2^1 \times 11^1 \times 281^1 \)[/tex]
### Part 1: Finding the smallest number whose prime factors are 2, 3, 5, and 11
To find the smallest number whose prime factors are 2, 3, 5, and 11, we simply multiply these prime numbers together:
[tex]\[ 2 \times 3 \times 5 \times 11 \][/tex]
Calculating the product:
[tex]\[ 2 \times 3 = 6 \][/tex]
[tex]\[ 6 \times 5 = 30 \][/tex]
[tex]\[ 30 \times 11 = 330 \][/tex]
So, the smallest number whose prime factors are 2, 3, 5, and 11 is:
Answer: (iv) 330
### Part 2: Prime Factorization of Given Numbers
#### (a) 90
Let's break down 90 using a prime factorization tree:
1. Divide by the smallest prime number (2):
[tex]\[ 90 \div 2 = 45 \][/tex]
2. Divide 45 by the next smallest prime number (3):
[tex]\[ 45 \div 3 = 15 \][/tex]
3. Continue with 3:
[tex]\[ 15 \div 3 = 5 \][/tex]
4. Finally, 5 is already a prime number:
[tex]\[ 5 \div 5 = 1 \][/tex]
So, the prime factorization of 90 is:
[tex]\[ 90 = 2 \times 3 \times 3 \times 5 \][/tex]
or in exponential form:
[tex]\[ 90 = 2^1 \times 3^2 \times 5^1 \][/tex]
#### (b) 749
1. Checking the divisibility by primes, you find that 749 is divisible by 7:
[tex]\[ 749 \div 7 = 107 \][/tex]
2. Next, checking 107, it's a prime number.
So, the prime factorization of 749 is:
[tex]\[ 749 = 7 \times 107 \][/tex]
#### (c) 625
1. Divide by the smallest prime number (5), since 625 ends with a 5:
[tex]\[ 625 \div 5 = 125 \][/tex]
2. Continue dividing by 5:
[tex]\[ 125 \div 5 = 25 \][/tex]
[tex]\[ 25 \div 5 = 5 \][/tex]
[tex]\[ 5 \div 5 = 1 \][/tex]
So, the prime factorization of 625 is:
[tex]\[ 625 = 5 \times 5 \times 5 \times 5 \][/tex]
or in exponential form:
[tex]\[ 625 = 5^4 \][/tex]
#### (d) 1480
1. Divide by the smallest prime number (2):
[tex]\[ 1480 \div 2 = 740 \][/tex]
[tex]\[ 740 \div 2 = 370 \][/tex]
[tex]\[ 370 \div 2 = 185 \][/tex]
2. Divide by the next smallest prime number (5):
[tex]\[ 185 \div 5 = 37 \][/tex]
3. Finally, 37 is a prime number.
So, the prime factorization of 1480 is:
[tex]\[ 1480 = 2 \times 2 \times 2 \times 5 \times 37 \][/tex]
or in exponential form:
[tex]\[ 1480 = 2^3 \times 5^1 \times 37^1 \][/tex]
#### (e) 6182
1. Divide by the smallest prime number (2):
[tex]\[ 6182 \div 2 = 3091 \][/tex]
2. Continue checking for prime factors:
3091 is not divisible by 2, 3, or 5. Checking for larger primes, we find:
[tex]\[ 3091 \div 11 = 281 \][/tex]
3. 281 is a prime number.
So, the prime factorization of 6182 is:
[tex]\[ 6182 = 2 \times 11 \times 281 \][/tex]
### Summary
- The smallest number whose prime factors are 2, 3, 5, and 11 is 330.
- Prime factorizations:
- 90: [tex]\( 2^1 \times 3^2 \times 5^1 \)[/tex]
- 749: [tex]\( 7^1 \times 107^1 \)[/tex]
- 625: [tex]\( 5^4 \)[/tex]
- 1480: [tex]\( 2^3 \times 5^1 \times 37^1 \)[/tex]
- 6182: [tex]\( 2^1 \times 11^1 \times 281^1 \)[/tex]
